Theorem undir 3747

Description: Distributive law for union over intersection. Theorem 29 of [Suppes] p. 27. (Contributed by NM, 30-Sep-2002.)

Assertion

Ref	Expression
undir	|- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )

Proof of Theorem undir

Step	Hyp	Ref	Expression
1		undi 3745	. 2 |- ( C u. ( A i^i B ) ) = ( ( C u. A ) i^i ( C u. B ) )
2		uncom 3648	. 2 |- ( ( A i^i B ) u. C ) = ( C u. ( A i^i B ) )
3		uncom 3648	. . 3 |- ( A u. C ) = ( C u. A )
4		uncom 3648	. . 3 |- ( B u. C ) = ( C u. B )
5	3, 4	ineq12i 3698	. 2 |- ( ( A u. C ) i^i ( B u. C ) ) = ( ( C u. A ) i^i ( C u. B ) )
6	1, 2, 5	3eqtr4i 2506	1 |- ( ( A i^i B ) u. C ) = ( ( A u. C ) i^i ( B u. C ) )

Syntax hints:   = wceq 1379   u. cun 3474   i^i cin 3475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-un 3481  df-in 3483

This theorem is referenced by:  undif1  3902  dfif4  3954  dfif5  3955  bwth  19676